Physical Chemistry Third Edition

660 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

We substitute (15.2-11) into Eq. (15.2-9) and divide byψ(x)η(t), obtaining

1

ψ

Hψ̂ ih ̄

η

dη

dt

(15.2-12)

The variablesxandtare now separated. Sincexandtare independent variables, each

can be held fixed while the other varies. Each side of the equation must be a constant

function of its argument and must be equal to the same constant, which we denote byE:

1

ψ

Hψ̂ E (15.2-13)

and

ih ̄

η

dη

dt

E (15.2-14)

Multiplication of the first equation byψand of the second equation byη/ih ̄gives

Hψ̂ Eψ (15.2-15)

and

dη

dt



E

ih ̄

η−

iE

h ̄

η (15.2-16)

Equation (15.2-15) is the same as the time-independent Schrödinger equation,

Eq. (15.2-8), soψis the coordinate wave function that satisfies that equation and

Eis the constant energy of the system.

Equation (15.2-16) has the solution

η(t)Ce−iEt/h ̄ (15.2-17)

whereCis a constant. Since the Eq. (15.2-16) is satisfied for any value ofCits value

is unimportant. We letC1 and write the complete wave function as

Ψ(x,t)ψ(x)e−iEt/h ̄ (15.2-18)

We will see that a solution to the time-independent Schrödinger equation provides

both a coordinate wave functionψand an energy valueE. We can immediately write a

solution to the time-dependent equation by multiplying a coordinate wave function by

the time factore−iEt/h ̄. This type of solution, with the coordinate and time dependence

in separate factors, corresponds to a standing wave, because any nodes are stationary.

There are also solutions of the time-dependent Schrödinger equation that are not prod-

ucts of a coordinate factor and a time factor. These solutions can correspond to traveling

waves.

The coordinate wave function can in many cases be chosen to be a real function.

The functionηis always complex, and can be written as a real part plus an imaginary

part by use of a mathematical identity (see Appendix B):

η(t)e−iEt/h ̄cos(−Et/h ̄)+isin(−Et/h ̄)cos(Et/h ̄)−isin(Et/h ̄) (15.2-19)